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Isaac Elishakoff, Vladik Kreinovich, Wolfram Luther, and Evgenija D. Popova 39 arithmetics like affine arithmetic or Taylor models to reduce the overestimation. Another way to tighten enclosures is to use contractors which identify parts of the decomposition disjoint with the object. In this talk, we will discuss how UniVerMeC helps us to integrate arbitrary interval contractors into the trees. This is a direct extension of our recent approach [4] which was limited formerly to implicit linear interval estimations [2]. The direct integration into the trees lets an algorithm take advantage of contractors’ properties without being aware of their actual use. This decoupling allows us to add or remove different contractors easily. In contrast to domain decompositions used, for example, in global optimization, interval trees have to cover the whole area by nodes properly. That is, we cannot dispose parts not containing the object but have to cover them with WHITE nodes indicating that they are empty. This makes employing interval contractors in trees complicated. We will show how to simplify this task by introducing special inversion nodes into the standard set. This does not change existing tree algorithms because nodes of the new type can be converted exactly into a set of standard nodes. References 1 G. Alefeld and J. Herzberger. Introduction to Interval Computations. Academic Press, 1983. 2 K. Bühler. Implicit linear interval estimations. In Proceedings of the 18th spring conference on Computer graphics, page 132. ACM, 2002. 3 E. Dyllong and S. Kiel. A comparison of verified distance computation between implicit objects using different arithmetics for range enclosure. In Proceedings of SCAN’2010. Submitted. 4 S. Kiel. Verified spatial subdivision of implicit objects using implicit linear interval estimations. In Curves and Surfaces 2011, volume 6920 of LNCS. Springer, 2011. To appear. 5 H Samet. Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann, San Francisco, 2006. 3.10 Degree-Based (Interval and Fuzzy) Techniques in Math and Science Education Olga M. Kosheleva (University of Texas – El Paso, US) License Creative Commons BY-NC-ND 3.0 Unported license © Olga M. Kosheleva In education, evaluations of the student’s knowledge, skills, and abilities are often subjective. Teachers and experts often make these evaluations by using words from natural language like “good”, “excellent”. Traditionally, in order to be able to process the evaluation results, these evaluations are first transformed into exact numbers. This transformation, however, ignores the uncertainty of the original estimates. To get a more adequate picture of the education process and education results, it is therefore desirable to transform these evaluations into intervals – or, more generally, fuzzy numbers. We show that this more adequate transformation can help on all the stages of the education process: in planning education, in teaching itself, and in assessing the education results. Specifically, in planning education and in teaching itself, interval and fuzzy techniques help us: 1 1 3 7 1
40 11371 – Uncertainty modeling and analysis with intervals: . . . better plan the order in which the material is presented and the amount of time allocated for each topic; interval and fuzzy techniques help us find the most efficient way of teaching interdisciplinary topics; these techniques also help to stimulate students by explaining historical (usually informal) motivations – often paradox-related motivations – behind different concepts and ideas of mathematics and science. In assessment, interval and fuzzy techniques help: to design a better grading scheme for test and assignments, a scheme that stimulates more effective learning, to provide a more adequate individual grading of contributions to group projects – by taking into account subjective estimates of different student distributions (and the uncertainty of these estimates), and to provide a more adequate description of the student knowledge and of the overall teaching effectiveness. The talk summarizes, combines, and expands on the ideas and results, some of which published in journals and conference proceedings. These published papers also contain additional technical details and practical examples of using these ideas. 3.11 A Comparison of Different Kinds of Multiple Precision and Arbitrary Precision Interval Arithmetics Walter Krämer (Universität Wuppertal, DE) License Creative Commons BY-NC-ND 3.0 Unported license © Walter Krämer Joint work of Krämer, Walter; Blomquist, Frithjof; Hofschuster, Werner URL http://www2.math.uni-wuppertal.de/org/WRST/index_de.html The current version of the C++ class library C-XSC for verified numerical computing offers quite a lot of different interval data types with different properties. We compared multiple precision data types like staggered precision data types (unevaluated sums of floatingpoint numbers) as well as arbitrary precision types based on arrays of integers and integer operations. In some respects staggered numbers and operation are restricted by properties of the underlying basic floating-point data type (IEEE double precision) whereas arbitrary precision numbers are only limited by the memory resources available. We presented some preliminary execution time comparisons and gave some advice when it is appropriate to use a specific multiple/arbitrary precision C-XSC data type. We also discussed the availability of some underlying external packages restricting the use of C- XSC’s arbitrary precision data types on some platforms. Several source code examples were presented to demonstrate the ease of use of the data types (due to operator and function name overloading) and the power of the different multiple/arbitrary precision C-XSC packages. Further features of C-XSC have been discussed in the talk “C-XSC – Overview and new developments” presented by Michael Zimmer during the course of this Dagstuhl seminar. Please refer to the corresponding abstract included in this document. Keywords: Multiple precision, arbitrary precision, staggered data types, C-XSC, MPFR, MPFI, interval computations.
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